Recall that for prime \(p\ge 5\)  and rational number \(a\in \mathbb {Z}_p\) , the \(p\) -adic Gamma function \(\Gamma _{p}\)  has the following basic properties [1]},
\(\Gamma _{p}(1)=-1,\quad \Gamma _{p}\left(\frac{1}{2}\right)^{2}=(-1)^{\frac{p+1}{2}}\quad (a)_n=(-1)^n\frac{\Gamma _{p}(a+n)}{\Gamma _{p}(a)},\quad G_1(a)=G_1(1-a),\) 
\(\Gamma _{p}(a+bp)\equiv \Gamma _{p}(a)\big (1+G_1(a)bp+\frac{G_2(a)b^2p^2}{2}\big ) \pmod {p^3},\) 
\(\Gamma _{p}(x) \Gamma _{p}(1-x)=(-1)^{a_{0}(x)},\) 
